Determine the value of the following complex number power. Your answer will be plotted in orange. $ ({\cos(\frac{19}{12}\pi) + i \sin(\frac{19}{12}\pi)}) ^ {8} $
Answer: Let's express our complex number in Euler form first. $ {\cos(\frac{19}{12}\pi) + i \sin(\frac{19}{12}\pi)} = { e^{19\pi i / 12}} $ Since $(a ^ b) ^ c = a ^ {b \cdot c}$ $ ({ e^{19\pi i / 12}}) ^ {8} = e ^ {8 \cdot (19\pi i / 12)} $ The angle of the result is $8 \cdot \frac{19}{12}\pi$ , which is $\frac{38}{3}\pi$ $\frac{38}{3}\pi$ is more than $2 \pi$ . It is a common practice to keep complex number angles between $0$ and $2 \pi$ , because $e^{2 \pi i} = (e^{\pi i}) ^ 2 = (-1) ^ 2 = 1$ . We will now subtract the nearest multiple of $2 \pi$ from the angle. $ \frac{38}{3}\pi - 12\pi = \frac{2}{3}\pi $ Our result is $ e^{2\pi i / 3}$. Converting this back from Euler form, we get $\cos(\frac{2}{3}\pi) + i \sin(\frac{2}{3}\pi)$.